If possible get hold of an old tire. Pump your tire up to P=400 kPa, drill a small hole, and monitor the pressure as it goes down. Predict what you think the graph of P against time t should look like. Discuss what mathematical form it should have.
The idea we develop is that the rate at which molecules escape from the tire is always proportional to the number in the tire. We can verify this by measuring the pressure at regular intervals and checking whether the change over the interval is proportional to the pressure at the start of the interval. From this we can get an exponential equation for P against time t.
The answer: Exponential decay. As seen in the above animation, the rate that air is escaping is proportional to the amount of air currently in the tire--resulting in an exponential decay function.The tire model is our introduction to exponential decay. This is a huge idea at this stage as it confronts the students with the two fundamental modes of change. Just as + and × are the two basic operations of arithmetic, so additive and multiplicative change are the basic modes of growth and decay. The most powerful way to think of these is in terms of rate of change. In additive growth, the rate of change is independent of size, whereas in multiplicative growth, the rate of change is proportional to size. This distinction will be encountered in different ways in the next units. Teacher manual
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